\(\int \frac {\cot ^3(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx\) [330]

Optimal result

Integrand size = 23, antiderivative size = 260 \[ \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} d}-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{4 (a-b)^{3/2} d}+\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{4 (a+b)^{3/2} d}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b} d}+\frac {\sqrt {a+b \sec (c+d x)}}{4 (a+b) d (1-\sec (c+d x))}+\frac {\sqrt {a+b \sec (c+d x)}}{4 (a-b) d (1+\sec (c+d x))} \]

[Out]

-1/4*b*arctanh((a+b*sec(d*x+c))^(1/2)/(a-b)^(1/2))/(a-b)^(3/2)/d+1/4*b*arctanh((a+b*sec(d*x+c))^(1/2)/(a+b)^(1
/2))/(a+b)^(3/2)/d-2*arctanh((a+b*sec(d*x+c))^(1/2)/a^(1/2))/d/a^(1/2)+arctanh((a+b*sec(d*x+c))^(1/2)/(a-b)^(1
/2))/d/(a-b)^(1/2)+arctanh((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2))/d/(a+b)^(1/2)+1/4*(a+b*sec(d*x+c))^(1/2)/(a+b)/
d/(1-sec(d*x+c))+1/4*(a+b*sec(d*x+c))^(1/2)/(a-b)/d/(1+sec(d*x+c))

Rubi [A] (verified)

Time = 0.34 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3970, 912, 1252, 212, 205, 213} \[ \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx=-\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{4 d (a-b)^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{d \sqrt {a-b}}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{d \sqrt {a+b}}+\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{4 d (a+b)^{3/2}}+\frac {\sqrt {a+b \sec (c+d x)}}{4 d (a+b) (1-\sec (c+d x))}+\frac {\sqrt {a+b \sec (c+d x)}}{4 d (a-b) (\sec (c+d x)+1)} \]

[In]

Int[Cot[c + d*x]^3/Sqrt[a + b*Sec[c + d*x]],x]

[Out]

(-2*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a]])/(Sqrt[a]*d) + ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]]/(Sq
rt[a - b]*d) - (b*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]])/(4*(a - b)^(3/2)*d) + (b*ArcTanh[Sqrt[a + b*S
ec[c + d*x]]/Sqrt[a + b]])/(4*(a + b)^(3/2)*d) + ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]]/(Sqrt[a + b]*d)
+ Sqrt[a + b*Sec[c + d*x]]/(4*(a + b)*d*(1 - Sec[c + d*x])) + Sqrt[a + b*Sec[c + d*x]]/(4*(a - b)*d*(1 + Sec[
c + d*x]))

Rule 205

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^n)^(p + 1)/(a*n*(p + 1))), x] + Dist[(n*(p
+ 1) + 1)/(a*n*(p + 1)), Int[(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[p, -1] && (
IntegerQ[2*p] || (n == 2 && IntegerQ[4*p]) || (n == 2 && IntegerQ[3*p]) || Denominator[p + 1/n] < Denominator[
p])

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 213

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[b, 2])^(-1))*ArcTanh[Rt[b, 2]*(x/Rt[-a, 2])]
, x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (LtQ[a, 0] || GtQ[b, 0])

Rule 912

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> With[{q = De
nominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + g*(x^q/e))^n*((c*d^2 + a*e^2)/e^2 - 2*c*
d*(x^q/e^2) + c*(x^(2*q)/e^2))^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*
g, 0] && NeQ[c*d^2 + a*e^2, 0] && IntegersQ[n, p] && FractionQ[m]

Rule 1252

Int[((d_) + (e_.)*(x_)^2)^(q_)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_), x_Symbol] :> Int[ExpandIntegrand[(d
+ e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b^2 - 4*a*c, 0] && ((Intege
rQ[p] && IntegerQ[q]) || IGtQ[p, 0] || IGtQ[q, 0])

Rule 3970

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Dist[-(-1)^((m - 1
)/2)/(d*b^(m - 1)), Subst[Int[(b^2 - x^2)^((m - 1)/2)*((a + x)^n/x), x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b
, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {b^4 \text {Subst}\left (\int \frac {1}{x \sqrt {a+x} \left (b^2-x^2\right )^2} \, dx,x,b \sec (c+d x)\right )}{d} \\ & = \frac {\left (2 b^4\right ) \text {Subst}\left (\int \frac {1}{\left (-a+x^2\right ) \left (-a^2+b^2+2 a x^2-x^4\right )^2} \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{d} \\ & = \frac {\left (2 b^4\right ) \text {Subst}\left (\int \left (-\frac {1}{b^4 \left (a-x^2\right )}+\frac {1}{4 b^3 \left (a+b-x^2\right )^2}+\frac {1}{2 b^4 \left (a+b-x^2\right )}-\frac {1}{4 b^3 \left (-a+b+x^2\right )^2}-\frac {1}{2 b^4 \left (-a+b+x^2\right )}\right ) \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{d}-\frac {\text {Subst}\left (\int \frac {1}{-a+b+x^2} \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{d}-\frac {2 \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{d}+\frac {b \text {Subst}\left (\int \frac {1}{\left (a+b-x^2\right )^2} \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{2 d}-\frac {b \text {Subst}\left (\int \frac {1}{\left (-a+b+x^2\right )^2} \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{2 d} \\ & = -\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} d}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b} d}+\frac {\sqrt {a+b \sec (c+d x)}}{4 (a+b) d (1-\sec (c+d x))}+\frac {\sqrt {a+b \sec (c+d x)}}{4 (a-b) d (1+\sec (c+d x))}+\frac {b \text {Subst}\left (\int \frac {1}{-a+b+x^2} \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{4 (a-b) d}+\frac {b \text {Subst}\left (\int \frac {1}{a+b-x^2} \, dx,x,\sqrt {a+b \sec (c+d x)}\right )}{4 (a+b) d} \\ & = -\frac {2 \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a} d}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b} d}-\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{4 (a-b)^{3/2} d}+\frac {b \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{4 (a+b)^{3/2} d}+\frac {\text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b} d}+\frac {\sqrt {a+b \sec (c+d x)}}{4 (a+b) d (1-\sec (c+d x))}+\frac {\sqrt {a+b \sec (c+d x)}}{4 (a-b) d (1+\sec (c+d x))} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.38 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.08 \[ \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx=\frac {-\frac {b^2 \arctan \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {-a+b}}\right )}{(-a+b)^{3/2}}-\frac {8 b \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a}}\right )}{\sqrt {a}}+\frac {4 b \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a-b}}\right )}{\sqrt {a-b}}+\frac {b^2 \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{(a+b)^{3/2}}-\frac {4 a \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )}{\sqrt {a+b}}+4 \sqrt {a+b} \text {arctanh}\left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )-\frac {b \sqrt {a+b \sec (c+d x)}}{(a+b) (-1+\sec (c+d x))}+\frac {b \sqrt {a+b \sec (c+d x)}}{(a-b) (1+\sec (c+d x))}}{4 b d} \]

[In]

Integrate[Cot[c + d*x]^3/Sqrt[a + b*Sec[c + d*x]],x]

[Out]

(-((b^2*ArcTan[Sqrt[a + b*Sec[c + d*x]]/Sqrt[-a + b]])/(-a + b)^(3/2)) - (8*b*ArcTanh[Sqrt[a + b*Sec[c + d*x]]
/Sqrt[a]])/Sqrt[a] + (4*b*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a - b]])/Sqrt[a - b] + (b^2*ArcTanh[Sqrt[a + b
*Sec[c + d*x]]/Sqrt[a + b]])/(a + b)^(3/2) - (4*a*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]])/Sqrt[a + b] +
4*Sqrt[a + b]*ArcTanh[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]] - (b*Sqrt[a + b*Sec[c + d*x]])/((a + b)*(-1 + Sec
[c + d*x])) + (b*Sqrt[a + b*Sec[c + d*x]])/((a - b)*(1 + Sec[c + d*x])))/(4*b*d)

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(3106\) vs. \(2(218)=436\).

Time = 2.01 (sec) , antiderivative size = 3107, normalized size of antiderivative = 11.95

method	result	size
default	\(\text {Expression too large to display}\)	\(3107\)

[In]

int(cot(d*x+c)^3/(a+b*sec(d*x+c))^(1/2),x,method=_RETURNVERBOSE)

[Out]

1/8/d/(a+b)^2/(a-b)^(5/2)/a*((a*(1-cos(d*x+c))^2*csc(d*x+c)^2-b*(1-cos(d*x+c))^2*csc(d*x+c)^2-a-b)/((1-cos(d*x
+c))^2*csc(d*x+c)^2-1))^(1/2)*((1-cos(d*x+c))^2*csc(d*x+c)^2-1)*(8*ln(2*(-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+b*
(1-cos(d*x+c))^2*csc(d*x+c)^2+2*a^(1/2)*((1-cos(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+c))^4*b*csc(d*x+c)^4-2*a*(
1-cos(d*x+c))^2*csc(d*x+c)^2+a+b)^(1/2)+2*a+b)/((1-cos(d*x+c))^2*csc(d*x+c)^2+1))*a^(7/2)*(a-b)^(3/2)*(1-cos(d
*x+c))^2*csc(d*x+c)^2+8*ln(2*(-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+b*(1-cos(d*x+c))^2*csc(d*x+c)^2+2*a^(1/2)*((1
-cos(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+c))^4*b*csc(d*x+c)^4-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+a+b)^(1/2)+2*a
+b)/((1-cos(d*x+c))^2*csc(d*x+c)^2+1))*a^(5/2)*(a-b)^(3/2)*b*(1-cos(d*x+c))^2*csc(d*x+c)^2-8*ln(2*(-2*a*(1-cos
(d*x+c))^2*csc(d*x+c)^2+b*(1-cos(d*x+c))^2*csc(d*x+c)^2+2*a^(1/2)*((1-cos(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+
c))^4*b*csc(d*x+c)^4-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+a+b)^(1/2)+2*a+b)/((1-cos(d*x+c))^2*csc(d*x+c)^2+1))*a^
(3/2)*(a-b)^(3/2)*b^2*(1-cos(d*x+c))^2*csc(d*x+c)^2-(a-b)^(3/2)*((1-cos(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+c)
)^4*b*csc(d*x+c)^4-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+a+b)^(1/2)*a^3*(1-cos(d*x+c))^4*csc(d*x+c)^4+2*(a-b)^(3/2
)*((1-cos(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+c))^4*b*csc(d*x+c)^4-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+a+b)^(1/2
)*a^2*b*(1-cos(d*x+c))^4*csc(d*x+c)^4-(a-b)^(3/2)*((1-cos(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+c))^4*b*csc(d*x+
c)^4-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+a+b)^(1/2)*a*b^2*(1-cos(d*x+c))^4*csc(d*x+c)^4-4*ln(2*(-a*(1-cos(d*x+c)
)^2*csc(d*x+c)^2+(a+b)^(1/2)*((1-cos(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+c))^4*b*csc(d*x+c)^4-2*a*(1-cos(d*x+c
))^2*csc(d*x+c)^2+a+b)^(1/2)+a+b)/(1-cos(d*x+c))^2*sin(d*x+c)^2)*(a+b)^(1/2)*(a-b)^(3/2)*a^3*(1-cos(d*x+c))^2*
csc(d*x+c)^2-ln(2*(-a*(1-cos(d*x+c))^2*csc(d*x+c)^2+(a+b)^(1/2)*((1-cos(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+c)
)^4*b*csc(d*x+c)^4-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+a+b)^(1/2)+a+b)/(1-cos(d*x+c))^2*sin(d*x+c)^2)*(a+b)^(1/2
)*(a-b)^(3/2)*a^2*b*(1-cos(d*x+c))^2*csc(d*x+c)^2+5*ln(2*(-a*(1-cos(d*x+c))^2*csc(d*x+c)^2+(a+b)^(1/2)*((1-cos
(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+c))^4*b*csc(d*x+c)^4-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+a+b)^(1/2)+a+b)/(1
-cos(d*x+c))^2*sin(d*x+c)^2)*(a+b)^(1/2)*(a-b)^(3/2)*a*b^2*(1-cos(d*x+c))^2*csc(d*x+c)^2-8*ln(2*(-2*a*(1-cos(d
*x+c))^2*csc(d*x+c)^2+b*(1-cos(d*x+c))^2*csc(d*x+c)^2+2*a^(1/2)*((1-cos(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+c)
)^4*b*csc(d*x+c)^4-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+a+b)^(1/2)+2*a+b)/((1-cos(d*x+c))^2*csc(d*x+c)^2+1))*a^(1
/2)*(a-b)^(3/2)*b^3*(1-cos(d*x+c))^2*csc(d*x+c)^2+(a-b)^(3/2)*((1-cos(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+c))^
4*b*csc(d*x+c)^4-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+a+b)^(1/2)*a^3*(1-cos(d*x+c))^2*csc(d*x+c)^2-4*(a-b)^(3/2)*
((1-cos(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+c))^4*b*csc(d*x+c)^4-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+a+b)^(1/2)*
a^2*b*(1-cos(d*x+c))^2*csc(d*x+c)^2-(a-b)^(3/2)*((1-cos(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+c))^4*b*csc(d*x+c)
^4-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+a+b)^(1/2)*a*b^2*(1-cos(d*x+c))^2*csc(d*x+c)^2+4*ln((a*(1-cos(d*x+c))^2*c
sc(d*x+c)^2-b*(1-cos(d*x+c))^2*csc(d*x+c)^2+((1-cos(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+c))^4*b*csc(d*x+c)^4-2
*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+a+b)^(1/2)*(a-b)^(1/2)-a)/(a-b)^(1/2))*a^5*(1-cos(d*x+c))^2*csc(d*x+c)^2-ln((
a*(1-cos(d*x+c))^2*csc(d*x+c)^2-b*(1-cos(d*x+c))^2*csc(d*x+c)^2+((1-cos(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+c)
)^4*b*csc(d*x+c)^4-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+a+b)^(1/2)*(a-b)^(1/2)-a)/(a-b)^(1/2))*a^4*(1-cos(d*x+c))
^2*b*csc(d*x+c)^2-9*ln((a*(1-cos(d*x+c))^2*csc(d*x+c)^2-b*(1-cos(d*x+c))^2*csc(d*x+c)^2+((1-cos(d*x+c))^4*a*cs
c(d*x+c)^4-(1-cos(d*x+c))^4*b*csc(d*x+c)^4-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+a+b)^(1/2)*(a-b)^(1/2)-a)/(a-b)^(
1/2))*a^3*b^2*(1-cos(d*x+c))^2*csc(d*x+c)^2+ln((a*(1-cos(d*x+c))^2*csc(d*x+c)^2-b*(1-cos(d*x+c))^2*csc(d*x+c)^
2+((1-cos(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+c))^4*b*csc(d*x+c)^4-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+a+b)^(1/2
)*(a-b)^(1/2)-a)/(a-b)^(1/2))*a^2*b^3*(1-cos(d*x+c))^2*csc(d*x+c)^2+5*ln((a*(1-cos(d*x+c))^2*csc(d*x+c)^2-b*(1
-cos(d*x+c))^2*csc(d*x+c)^2+((1-cos(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+c))^4*b*csc(d*x+c)^4-2*a*(1-cos(d*x+c)
)^2*csc(d*x+c)^2+a+b)^(1/2)*(a-b)^(1/2)-a)/(a-b)^(1/2))*a*b^4*(1-cos(d*x+c))^2*csc(d*x+c)^2+(a-b)^(3/2)*((1-co
s(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+c))^4*b*csc(d*x+c)^4-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+a+b)^(3/2)*a^2-(a
-b)^(3/2)*((1-cos(d*x+c))^4*a*csc(d*x+c)^4-(1-cos(d*x+c))^4*b*csc(d*x+c)^4-2*a*(1-cos(d*x+c))^2*csc(d*x+c)^2+a
+b)^(3/2)*a*b)/((a*(1-cos(d*x+c))^2*csc(d*x+c)^2-b*(1-cos(d*x+c))^2*csc(d*x+c)^2-a-b)*((1-cos(d*x+c))^2*csc(d*
x+c)^2-1))^(1/2)/(1-cos(d*x+c))^2*sin(d*x+c)^2

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 466 vs. \(2 (216) = 432\).

Time = 25.89 (sec) , antiderivative size = 4336, normalized size of antiderivative = 16.68 \[ \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx=\text {Too large to display} \]

[In]

integrate(cot(d*x+c)^3/(a+b*sec(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

[-1/16*(8*(a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(a)*log(-8*a^2*cos(d*x + c)^2 -
8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x + c) + b)/cos(d*x
+ c))) + (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3 - (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*cos(d*x + c)^2)*sqrt
(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(
a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x +
c) + 1)) + (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 - (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3)*cos(d*x + c)^2)*s
qrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sq
rt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*
x + c) + 1)) - 8*((a^4 - a^2*b^2)*cos(d*x + c)^2 - (a^3*b - a*b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos
(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*d), -1/16*(16*(a^4 - 2*a^2
*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x + c) + b)/cos(
d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) + (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3 - (4*a^4 + 3*a^3*b - 6
*a^2*b^2 - 5*a*b^3)*cos(d*x + c)^2)*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a - b
)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos
(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) + (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 - (4*a^4 - 3*a^3*b
- 6*a^2*b^2 + 5*a*b^3)*cos(d*x + c)^2)*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a
+ b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*
cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)) - 8*((a^4 - a^2*b^2)*cos(d*x + c)^2 - (a^3*b - a*b^3)*cos
(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (a^5 - 2*a^3
*b^2 + a*b^4)*d), -1/16*(2*(4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3 - (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*co
s(d*x + c)^2)*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a -
b)*cos(d*x + c) + b)) + 8*(a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(a)*log(-8*a^2
*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*cos(d*x +
c) + b)/cos(d*x + c))) + (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 - (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3)*cos
(d*x + c)^2)*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a + b)*cos(d*x + c)^2 + b*co
s(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b + 3*b^2)*cos(d*x + c))/(cos(d*x + c
)^2 - 2*cos(d*x + c) + 1)) - 8*((a^4 - a^2*b^2)*cos(d*x + c)^2 - (a^3*b - a*b^3)*cos(d*x + c))*sqrt((a*cos(d*x
+ c) + b)/cos(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*d), -1/16*(1
6*(a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*cos(d*x
+ c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) + 2*(4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3 - (4*a
^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*cos(d*x + c)^2)*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) +
b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) + (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 - (4*a^4
- 3*a^3*b - 6*a^2*b^2 + 5*a*b^3)*cos(d*x + c)^2)*sqrt(a + b)*log(-((8*a^2 + 8*a*b + b^2)*cos(d*x + c)^2 + b^2
+ 4*((2*a + b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*
b + 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 - 2*cos(d*x + c) + 1)) - 8*((a^4 - a^2*b^2)*cos(d*x + c)^2 - (a^3*b -
a*b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (
a^5 - 2*a^3*b^2 + a*b^4)*d), 1/16*(2*(4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 - (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5
*a*b^3)*cos(d*x + c)^2)*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c
)/((2*a + b)*cos(d*x + c) + b)) - 8*(a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(a)*l
og(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt((a*
cos(d*x + c) + b)/cos(d*x + c))) - (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3 - (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a
*b^3)*cos(d*x + c)^2)*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)^2 + b^2 + 4*((2*a - b)*cos(d*x + c)
^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) + 2*(4*a*b - 3*b^2)*cos(d*x + c))/(co
s(d*x + c)^2 + 2*cos(d*x + c) + 1)) + 8*((a^4 - a^2*b^2)*cos(d*x + c)^2 - (a^3*b - a*b^3)*cos(d*x + c))*sqrt((
a*cos(d*x + c) + b)/cos(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*d),
-1/16*(16*(a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a
*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) - 2*(4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b
^3 - (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3)*cos(d*x + c)^2)*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*
x + c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) + (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3
- (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*cos(d*x + c)^2)*sqrt(a - b)*log(-((8*a^2 - 8*a*b + b^2)*cos(d*x + c)
^2 + b^2 + 4*((2*a - b)*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)) +
2*(4*a*b - 3*b^2)*cos(d*x + c))/(cos(d*x + c)^2 + 2*cos(d*x + c) + 1)) - 8*((a^4 - a^2*b^2)*cos(d*x + c)^2 -
(a^3*b - a*b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x +
c)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*d), -1/8*((4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3 - (4*a^4 + 3*a^3*b - 6*a^2*b
^2 - 5*a*b^3)*cos(d*x + c)^2)*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(
d*x + c)/((2*a - b)*cos(d*x + c) + b)) - (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 - (4*a^4 - 3*a^3*b - 6*a^2*b^2
+ 5*a*b^3)*cos(d*x + c)^2)*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c) + b)/cos(d*x + c))*cos(d*x
+ c)/((2*a + b)*cos(d*x + c) + b)) + 4*(a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(
a)*log(-8*a^2*cos(d*x + c)^2 - 8*a*b*cos(d*x + c) - b^2 + 4*(2*a*cos(d*x + c)^2 + b*cos(d*x + c))*sqrt(a)*sqrt
((a*cos(d*x + c) + b)/cos(d*x + c))) - 4*((a^4 - a^2*b^2)*cos(d*x + c)^2 - (a^3*b - a*b^3)*cos(d*x + c))*sqrt(
(a*cos(d*x + c) + b)/cos(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (a^5 - 2*a^3*b^2 + a*b^4)*d)
, -1/8*(8*(a^4 - 2*a^2*b^2 + b^4 - (a^4 - 2*a^2*b^2 + b^4)*cos(d*x + c)^2)*sqrt(-a)*arctan(2*sqrt(-a)*sqrt((a*
cos(d*x + c) + b)/cos(d*x + c))*cos(d*x + c)/(2*a*cos(d*x + c) + b)) + (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3
- (4*a^4 + 3*a^3*b - 6*a^2*b^2 - 5*a*b^3)*cos(d*x + c)^2)*sqrt(-a + b)*arctan(-2*sqrt(-a + b)*sqrt((a*cos(d*x
+ c) + b)/cos(d*x + c))*cos(d*x + c)/((2*a - b)*cos(d*x + c) + b)) - (4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3 -
(4*a^4 - 3*a^3*b - 6*a^2*b^2 + 5*a*b^3)*cos(d*x + c)^2)*sqrt(-a - b)*arctan(2*sqrt(-a - b)*sqrt((a*cos(d*x + c
) + b)/cos(d*x + c))*cos(d*x + c)/((2*a + b)*cos(d*x + c) + b)) - 4*((a^4 - a^2*b^2)*cos(d*x + c)^2 - (a^3*b -
a*b^3)*cos(d*x + c))*sqrt((a*cos(d*x + c) + b)/cos(d*x + c)))/((a^5 - 2*a^3*b^2 + a*b^4)*d*cos(d*x + c)^2 - (
a^5 - 2*a^3*b^2 + a*b^4)*d)]

Sympy [F]

\[ \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx=\int \frac {\cot ^{3}{\left (c + d x \right )}}{\sqrt {a + b \sec {\left (c + d x \right )}}}\, dx \]

[In]

integrate(cot(d*x+c)**3/(a+b*sec(d*x+c))**(1/2),x)

[Out]

Integral(cot(c + d*x)**3/sqrt(a + b*sec(c + d*x)), x)

Maxima [F]

\[ \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx=\int { \frac {\cot \left (d x + c\right )^{3}}{\sqrt {b \sec \left (d x + c\right ) + a}} \,d x } \]

[In]

integrate(cot(d*x+c)^3/(a+b*sec(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(cot(d*x + c)^3/sqrt(b*sec(d*x + c) + a), x)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 533 vs. \(2 (216) = 432\).

Time = 0.92 (sec) , antiderivative size = 533, normalized size of antiderivative = 2.05 \[ \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx=\frac {\frac {16 \, \arctan \left (-\frac {\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b} + \sqrt {a - b}}{2 \, \sqrt {-a}}\right )}{\sqrt {-a}} - \frac {2 \, {\left (4 \, a + 5 \, b\right )} \arctan \left (-\frac {\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b}}{\sqrt {-a - b}}\right )}{{\left (a + b\right )} \sqrt {-a - b}} + \frac {{\left (4 \, a - 5 \, b\right )} \log \left ({\left | {\left (\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b}\right )} {\left (a - b\right )} - \sqrt {a - b} a \right |}\right )}{{\left (a - b\right )}^{\frac {3}{2}}} + \frac {\sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b}}{a - b} - \frac {2 \, {\left ({\left (\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b}\right )} a - {\left (a + b\right )} \sqrt {a - b}\right )}}{{\left ({\left (\sqrt {a - b} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - \sqrt {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 2 \, a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b}\right )}^{2} - a - b\right )} {\left (a + b\right )}}}{8 \, d \mathrm {sgn}\left (\cos \left (d x + c\right )\right )} \]

[In]

integrate(cot(d*x+c)^3/(a+b*sec(d*x+c))^(1/2),x, algorithm="giac")

[Out]

1/8*(16*arctan(-1/2*(sqrt(a - b)*tan(1/2*d*x + 1/2*c)^2 - sqrt(a*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*
c)^4 - 2*a*tan(1/2*d*x + 1/2*c)^2 + a + b) + sqrt(a - b))/sqrt(-a))/sqrt(-a) - 2*(4*a + 5*b)*arctan(-(sqrt(a -
b)*tan(1/2*d*x + 1/2*c)^2 - sqrt(a*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 - 2*a*tan(1/2*d*x + 1/2*
c)^2 + a + b))/sqrt(-a - b))/((a + b)*sqrt(-a - b)) + (4*a - 5*b)*log(abs((sqrt(a - b)*tan(1/2*d*x + 1/2*c)^2
- sqrt(a*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 - 2*a*tan(1/2*d*x + 1/2*c)^2 + a + b))*(a - b) - sq
rt(a - b)*a))/(a - b)^(3/2) + sqrt(a*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 - 2*a*tan(1/2*d*x + 1/2
*c)^2 + a + b)/(a - b) - 2*((sqrt(a - b)*tan(1/2*d*x + 1/2*c)^2 - sqrt(a*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*
x + 1/2*c)^4 - 2*a*tan(1/2*d*x + 1/2*c)^2 + a + b))*a - (a + b)*sqrt(a - b))/(((sqrt(a - b)*tan(1/2*d*x + 1/2*
c)^2 - sqrt(a*tan(1/2*d*x + 1/2*c)^4 - b*tan(1/2*d*x + 1/2*c)^4 - 2*a*tan(1/2*d*x + 1/2*c)^2 + a + b))^2 - a -
b)*(a + b)))/(d*sgn(cos(d*x + c)))

Mupad [F(-1)]

Timed out. \[ \int \frac {\cot ^3(c+d x)}{\sqrt {a+b \sec (c+d x)}} \, dx=\int \frac {{\mathrm {cot}\left (c+d\,x\right )}^3}{\sqrt {a+\frac {b}{\cos \left (c+d\,x\right )}}} \,d x \]

[In]

int(cot(c + d*x)^3/(a + b/cos(c + d*x))^(1/2),x)

[Out]

int(cot(c + d*x)^3/(a + b/cos(c + d*x))^(1/2), x)